Abstract
We provide a recursive formula for the computation of moments of distributions belonging to a subclass of the exponential family. This subclass includes important cases as the binomial, negative binomial, Poisson, gamma and normal distribution, among others. The recursive formula provides a procedure to sequentially calculate the moments using only elementary operations. The approach makes no use of the moment generating function.
References
N.I. Akhiezer, The classical moment problem and some related questions in analysis, Oliver & Boyd, Edinburgh, 1965.
A. Bényi, S. M. Manago, A recursive formula for moments of a binomial distribution, The College Mathematics Journal, 36(2005), no. 1, 68–72. Doi: 10.2307/30044825
B. Chan, Derivation of moment formulas by operator valued probability generating functions, The American Statistician 36(1982), no. 3a, 179–181. Doi:10.1080/00031305.1982.10482826.
H. Cramér, Mathematical Methods of Statistics. Princeton University Press, Princeton NJ, 1946. Doi: 10.1515/9781400883868
H.M. Hudson, A natural identity for exponential families with applications in multiparameter estimation, The Annals of Statistics, 6(1978), no. 3, 473–484. Doi: 10.1214/aos/1176344194
T. Jin, S.B. Provost, J. Ren, Moment-based density approximations for aggregate losses, Scandinavian Actuarial Journal, (2016), no. 3, 216–245. Doi:10.1080/03461238.2014.921640
J.S.J. Kang, S.B. Provost, J. Ren, Moment-based density approximation techniques as applied to heavy tailed distributions, International Journal of Statistics and Probability, 8(2019), no. 3. Doi:10.5539/ijsp.v8n3p1
S.K. Kattumannil, On Stein’s identity and its applications, Statistics and Probability Letters, 79 (2009) 1444 1449. Doi: 10.1016/j.spl.2009.03.021
M.G. Kendall, The Advanced Theory of Statistics. Charles Griffin & Company, London, 1945.
E.L. Lehmann, G. Casella, Theory of Point Estimation, 2nd edition, Springer, New York NY, 1988. Doi: 10.1007/b98854
F.R. Link, Moments of discrete probability distributions derived using finite difference operators, The American Statistician, 35(1981), no.1, 44–46. Doi:10.2307/2683584
N. Mukhopadhyay, Introductory Statistical Inference, Chapman and Hall/CRC, New York NY, 2006. Doi: 10.1201/b16497
J. Munkhammar, L. Mattsson, J. Rydén, Polynomial probability distribution estimation using the method of moments, PLOS ONE, 12(2017), no.4. Doi: 10.1371/journal.pone.0174573
S. Nadarajah, J. Chu, X. Jiang, On moment based density approximations for aggregate losses, Journal of Computational and Applied Mathematics, 298(2016), 152–166. Doi:10.1016/j.cam.2015.11.048
C. Philipson, A note on moments of a Poisson probability distribution, Scandinavian Actuarial Journal, (1963), no.3-4, 243-244. Doi: 10.1080/03461238.1963.10410613
S.B. Provost, Moment-based density approximants, The Mathematica Journal, 9(2005), no.4, 727–756.
SageMath, Open-source Mathematics Software System. In: https://www.sagemath.org. Accessed by 10/07/2020.
C.M. Stein, Estimation of the mean of a multivariate normal distribution, The Annals of Statistics 9(1981), no. 6, 1135–1151. Doi: 10.1214/aos/1176345632
B. Sundt, W.S. Jewell, Further results on recursive evaluation of compound distributions, ASTIN Bulletin: The Journal of the International Actuarial Association 12(1981), no.1, 27–39. Doi: doi:10.1017/ S0515036100006802
Comments
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Copyright (c) 2021 Revista de Matemática: Teoría y Aplicaciones